Subgroups — C₅×#63881 #13087165

Order 1440.

430 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#626	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×D₃×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3722	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×Dic₃²	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#893	—	3	1	no	maximal
9	C₅×#67⋊C₄	—	3	1	no	maximal
10	C₂²×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₄×C₅×C₃⋊D₃	—	2	2	no	—
15	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₅×#626	C₄	1	4	yes	complemented
19	C₅×D₃×Dic₃	—	2	2	no	—
20	#63881	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×#67	—	3	1	no	—
22	C₂²×C₅×Dic₃	—	3	1	no	—
23	C₂×C₅×D₁₂	—	3	1	no	—
24	C₂×C₄×C₅×D₃	—	3	1	no	—
25	C₂×C₄×C₅×D₃	—	3	1	no	—
26	C₂×C₄×C₅×D₃	—	3	1	no	—
27	C₂×C₄×C₅×D₃	—	6	1	no	—
28	C₃×C₅×C₄⋊C₄	—	3	1	no	—
29	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
30	C₅×(C₂×D₃)⋊C₄	—	3	1	no	—
31	C₅×(C₂×D₃)⋊C₄	—	3	1	no	—
32	C₅×(C₂×D₃)⋊C₄	D₃	1	1	yes	characteristic, complemented
33	C₅×Dic₃⋊C₄	D₃	1	1	yes	characteristic, complemented
34	C₄×C₅×Dic₃	—	3	1	no	—
35	C₄×C₅×Dic₃	—	3	1	no	—
36	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₅×C₃⋊D₃	—	2	1	no	—
38	C₂×C₅×C₃⋊D₃	—	2	1	no	—
39	C₂×C₅×C₃⋊D₃	C₂×C₄	1	2	yes	complemented
40	C₂×C₃×C₅×D₃	—	2	1	no	—
41	C₂×C₃×C₅×D₃	—	2	1	no	—
42	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	complemented
43	C₃²×C₄×C₅	—	2	1	no	—
44	C₅×C₃⋊Dic₃	D₄	1	2	yes	complemented
45	C₃×C₅×Dic₃	—	2	1	no	—
46	C₃×C₅×Dic₃	—	2	1	no	—
47	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	complemented
48	C₄×C₅×D₄	—	9	1	no	Hall
49	C₂×C₄×C₃⋊D₃	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×#626	C₂×C₅	1	1	yes	characteristic
51	C₂×D₃×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
52	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
54	#3722	C₂×C₅	1	1	yes	characteristic, complemented
55	Dic₃²	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂³×C₃×C₅	—	3	1	no	—
57	C₂²×C₅×D₃	—	3	1	no	—
58	C₂²×C₅×D₃	—	3	1	no	—
59	C₂²×C₅×D₃	—	6	1	no	—
60	C₂²×C₅×D₃	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	3	1	no	—
63	C₂×C₃×C₄×C₅	—	3	1	no	—
64	C₂×C₃×C₄×C₅	—	3	1	no	—
65	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
67	C₅×#67	—	3	4	no	—
68	C₂×C₅×Dic₃	—	3	1	no	—
69	C₂×C₅×Dic₃	—	3	1	no	—
70	C₂×C₅×Dic₃	—	6	2	no	—
71	C₂×C₅×Dic₃	—	6	1	no	—
72	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
73	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
74	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
75	C₅×D₁₂	—	3	4	no	—
76	C₄×C₅×D₃	—	6	2	no	—
77	C₄×C₅×D₃	—	6	2	no	—
78	C₄×C₅×D₃	—	6	2	no	—
79	C₄×C₅×D₃	—	6	4	no	—
80	#893	—	3	1	no	—
81	#67⋊C₄	—	3	1	no	—
82	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₅	#41	1	1	yes	characteristic
85	C₅×C₃⋊D₃	—	2	2	no	—
86	C₃×C₅×D₃	—	2	2	no	—
87	C₂×C₅×D₄	—	9	1	no	—
88	C₂²×C₄×C₅	—	9	1	no	—
89	C₂²×C₄×C₅	—	9	1	no	—
90	C₅×C₄⋊C₄	—	3	1	no	—
91	C₅×C₂²⋊C₄	—	3	1	no	—
92	C₅×C₂²⋊C₄	—	9	1	no	—
93	C₄²×C₅	—	9	1	no	—
94	C₂²×C₃⋊D₃	C₂²×C₅	1	1	yes	characteristic
95	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₃⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
98	C₄×C₃⋊D₃	—	2	2	no	—
99	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
102	#626	C₄×C₅	1	4	yes	complemented
103	D₃×Dic₃	—	2	2	no	—
104	C₂²×C₃×C₅	—	2	1	no	—
105	C₂²×C₃×C₅	—	3	2	no	—
106	C₂²×C₃×C₅	—	6	1	no	—
107	C₂²×C₃×C₅	—	6	1	no	—
108	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
109	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₅×D₃	—	2	1	no	—
111	C₂×C₅×D₃	—	2	1	no	—
112	C₂×C₅×D₃	—	3	2	no	—
113	C₂×C₅×D₃	—	3	2	no	—
114	C₂×C₅×D₃	—	6	2	no	—
115	C₂×C₅×D₃	—	6	2	no	—
116	C₂×C₅×D₃	—	6	2	no	—
117	C₂×C₅×D₃	—	6	1	no	—
118	C₂×C₅×D₃	—	6	1	no	—
119	C₂×C₅×D₃	—	6	1	no	—
120	C₂×C₅×D₃	—	6	1	no	—
121	C₂×C₅×D₃	C₄×D₃	1	2	yes	complemented
122	C₃×C₄×C₅	—	2	2	no	—
123	C₃×C₄×C₅	—	2	1	no	—
124	C₃×C₄×C₅	—	2	1	no	—
125	C₃×C₄×C₅	—	3	2	no	—
126	C₃×C₄×C₅	—	6	1	no	—
127	C₃×C₄×C₅	—	6	1	no	—
128	C₅×Dic₃	—	2	1	no	—
129	C₅×Dic₃	—	2	1	no	—
130	C₅×Dic₃	—	3	2	no	—
131	C₅×Dic₃	—	3	2	no	—
132	C₅×Dic₃	—	6	2	no	—
133	C₅×Dic₃	C₄×D₃	1	2	yes	complemented
134	C₂×#67	—	3	1	no	—
135	C₂²×Dic₃	—	3	1	no	—
136	C₂×D₁₂	—	3	1	no	—
137	C₂×C₄×D₃	—	3	1	no	—
138	C₂×C₄×D₃	—	3	1	no	—
139	C₂×C₄×D₃	—	3	1	no	—
140	C₂×C₄×D₃	—	6	1	no	—
141	C₃×C₄⋊C₄	—	3	1	no	—
142	C₃×C₂²⋊C₄	—	3	1	no	—
143	(C₂×D₃)⋊C₄	—	3	1	no	—
144	(C₂×D₃)⋊C₄	—	3	1	no	—
145	(C₂×D₃)⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
146	Dic₃⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
147	C₄×Dic₃	—	3	1	no	—
148	C₄×Dic₃	—	3	1	no	—
149	C₃²×C₅	C₄×D₄	1	1	yes	characteristic, complemented, Hall
150	C₂³×C₅	—	3	1	no	—
151	C₂³×C₅	—	9	1	no	—
152	C₅×D₄	—	9	4	no	—
153	C₂×C₄×C₅	—	3	1	no	—
154	C₂×C₄×C₅	—	3	1	no	—
155	C₂×C₄×C₅	—	3	1	no	—
156	C₂×C₄×C₅	—	9	1	no	—
157	C₂×C₄×C₅	—	18	2	no	—
158	C₂×C₄×C₅	—	18	2	no	—
159	C₂×C₄×C₅	D₃²	1	1	yes	characteristic
160	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₃⋊D₃	—	2	1	no	—
162	C₂×C₃⋊D₃	—	2	1	no	—
163	C₂×C₃⋊D₃	C₂×C₄×C₅	1	2	yes	complemented
164	C₂×C₃×D₃	—	2	1	no	—
165	C₂×C₃×D₃	—	2	1	no	—
166	C₂×C₃×D₃	C₂×C₄×C₅	1	2	yes	complemented
167	C₃²×C₄	—	2	1	no	—
168	C₃⋊Dic₃	C₅×D₄	1	2	yes	complemented
169	C₃×Dic₃	—	2	1	no	—
170	C₃×Dic₃	—	2	1	no	—
171	C₃×Dic₃	C₂×C₄×C₅	1	2	yes	complemented
172	C₄×D₄	—	9	1	no	Sylow
173	C₂×C₃×C₅	—	2	1	no	—
174	C₂×C₃×C₅	—	2	1	no	—
175	C₂×C₃×C₅	—	2	1	no	—
176	C₂×C₃×C₅	—	6	2	no	—
177	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
180	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
181	C₂×C₃×C₅	#237	1	1	yes	characteristic
182	C₂×C₃×C₅	#239	1	1	yes	characteristic
183	C₅×D₃	—	2	2	no	—
184	C₅×D₃	—	6	4	no	—
185	C₅×D₃	—	6	2	no	—
186	C₅×D₃	—	6	2	no	—
187	C₂³×C₃	—	3	1	no	—
188	C₂²×D₃	—	3	1	no	—
189	C₂²×D₃	—	3	1	no	—
190	C₂²×D₃	—	6	1	no	—
191	C₂²×D₃	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	3	1	no	—
194	C₂×C₃×C₄	—	3	1	no	—
195	C₂×C₃×C₄	—	3	1	no	—
196	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
198	#67	—	3	4	no	—
199	C₂×Dic₃	—	3	1	no	—
200	C₂×Dic₃	—	3	1	no	—
201	C₂×Dic₃	—	6	1	no	—
202	C₂×Dic₃	—	6	2	no	—
203	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
204	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
205	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
206	D₁₂	—	3	4	no	—
207	C₄×D₃	—	6	2	no	—
208	C₄×D₃	—	6	2	no	—
209	C₄×D₃	—	6	2	no	—
210	C₄×D₃	—	6	4	no	—
211	C₂²×C₅	—	3	2	no	—
212	C₂²×C₅	—	6	1	no	—
213	C₂²×C₅	—	6	1	no	—
214	C₂²×C₅	—	9	2	no	—
215	C₂²×C₅	—	18	1	no	—
216	C₂²×C₅	—	18	1	no	—
217	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	3	2	no	—
220	C₄×C₅	—	6	1	no	—
221	C₄×C₅	—	6	1	no	—
222	C₄×C₅	—	9	2	no	—
223	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₃²	C₅×#41	1	1	yes	characteristic
226	C₃⋊D₃	—	2	2	no	—
227	C₃×D₃	—	2	2	no	—
228	C₂×D₄	—	9	1	no	—
229	C₂²×C₄	—	9	1	no	—
230	C₂²×C₄	—	9	1	no	—
231	C₄⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	3	1	no	—
233	C₂²⋊C₄	—	9	1	no	—
234	C₄²	—	9	1	no	—
235	C₃×C₅	—	2	1	no	—
236	C₃×C₅	#67⋊C₄	1	1	yes	characteristic, complemented
237	C₃×C₅	#893	1	1	yes	characteristic, complemented
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	3	2	no	—
240	C₂²×C₃	—	6	1	no	—
241	C₂²×C₃	—	6	1	no	—
242	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
243	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
244	C₂×D₃	—	2	1	no	—
245	C₂×D₃	—	2	1	no	—
246	C₂×D₃	—	3	2	no	—
247	C₂×D₃	—	3	2	no	—
248	C₂×D₃	—	6	2	no	—
249	C₂×D₃	—	6	2	no	—
250	C₂×D₃	—	6	2	no	—
251	C₂×D₃	—	6	1	no	—
252	C₂×D₃	—	6	1	no	—
253	C₂×D₃	—	6	1	no	—
254	C₂×D₃	—	6	1	no	—
255	C₂×D₃	C₄×C₅×D₃	1	2	yes	complemented
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	3	2	no	—
260	C₃×C₄	—	6	1	no	—
261	C₃×C₄	—	6	1	no	—
262	Dic₃	—	2	1	no	—
263	Dic₃	—	2	1	no	—
264	Dic₃	—	3	2	no	—
265	Dic₃	—	3	2	no	—
266	Dic₃	—	6	2	no	—
267	Dic₃	C₄×C₅×D₃	1	2	yes	complemented
268	C₂×C₅	—	6	2	no	—
269	C₂×C₅	—	18	2	no	—
270	C₂×C₅	#3796	1	1	yes	characteristic
271	C₂×C₅	C₄×D₃²	1	1	yes	characteristic
272	C₂×C₅	#3811	1	1	yes	characteristic
273	C₃²	C₄×C₅×D₄	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	3	1	no	—
275	C₂³	—	9	1	no	—
276	D₄	—	9	4	no	—
277	C₂×C₄	—	3	1	no	—
278	C₂×C₄	—	3	1	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	9	1	no	—
281	C₂×C₄	—	18	2	no	—
282	C₂×C₄	—	18	2	no	—
283	C₂×C₄	C₅×D₃²	1	1	yes	characteristic
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	6	2	no	—
288	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
289	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
290	C₂×C₃	C₅×D₃×D₄	1	1	yes	characteristic
291	C₂×C₃	C₅×D₃×D₄	1	1	yes	characteristic
292	C₂×C₃	C₅×#237	1	1	yes	characteristic
293	C₂×C₃	C₅×#239	1	1	yes	characteristic
294	D₃	—	2	2	no	—
295	D₃	—	6	4	no	—
296	D₃	—	6	2	no	—
297	D₃	—	6	2	no	—
298	C₅	#63881	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	3	2	no	—
300	C₂²	—	6	1	no	—
301	C₂²	—	6	1	no	—
302	C₂²	—	9	2	no	—
303	C₂²	—	18	1	no	—
304	C₂²	—	18	1	no	—
305	C₂²	C₂×C₅×D₃²	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	3	2	no	—
308	C₄	—	6	1	no	—
309	C₄	—	6	1	no	—
310	C₄	—	9	2	no	—
311	C₃	—	2	1	no	—
312	C₃	C₅×#67⋊C₄	1	1	yes	characteristic, complemented
313	C₃	C₅×#893	1	1	yes	characteristic, complemented
314	C₂	—	6	2	no	—
315	C₂	—	18	2	no	—
316	C₂	C₅×#3796	1	1	yes	characteristic
317	C₂	C₄×C₅×D₃²	1	1	yes	characteristic
318	C₂	C₅×#3811	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₄×C₅
Frattini	C₂²	C₂×C₅×D₃²
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂×C₅×#626	C₂
cosocle	C₂²×C₃²	C₂³×C₅
hypercenter	C₂×C₄×C₅	D₃²
hypocenter	C₃²	C₄×C₅×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₄×C₅×C₃⋊D₃	C₂	semidirect
3. C₂×C₅×D₃×Dic₃	C₂	semidirect
4. C₃×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
5. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
6. C₅×#3722	C₂	semidirect
7. C₅×Dic₃²	C₂	semidirect
18. C₅×#626	C₄	semidirect
20. #63881	C₅	direct (N × H)
32. C₅×(C₂×D₃)⋊C₄	D₃	semidirect
33. C₅×Dic₃⋊C₄	D₃	semidirect
39. C₂×C₅×C₃⋊D₃	C₂×C₄	semidirect
42. C₂×C₃×C₅×D₃	C₂×C₄	semidirect
44. C₅×C₃⋊Dic₃	D₄	semidirect
47. C₃×C₅×Dic₃	C₂×C₄	semidirect
49. C₂×C₄×C₃⋊D₃	C₂×C₅	semidirect
51. C₂×D₃×Dic₃	C₂×C₅	semidirect
52. C₃×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
53. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
54. #3722	C₂×C₅	semidirect
55. Dic₃²	C₂×C₅	semidirect
102. #626	C₄×C₅	semidirect
121. C₂×C₅×D₃	C₄×D₃	semidirect
133. C₅×Dic₃	C₄×D₃	semidirect
145. (C₂×D₃)⋊C₄	C₅×D₃	semidirect
146. Dic₃⋊C₄	C₅×D₃	semidirect
149. C₃²×C₅	C₄×D₄	semidirect
163. C₂×C₃⋊D₃	C₂×C₄×C₅	semidirect
166. C₂×C₃×D₃	C₂×C₄×C₅	semidirect
168. C₃⋊Dic₃	C₅×D₄	semidirect
171. C₃×Dic₃	C₂×C₄×C₅	semidirect
236. C₃×C₅	#67⋊C₄	semidirect
237. C₃×C₅	#893	semidirect
255. C₂×D₃	C₄×C₅×D₃	semidirect
267. Dic₃	C₄×C₅×D₃	semidirect
273. C₃²	C₄×C₅×D₄	semidirect
298. C₅	#63881	direct (N × H)
312. C₃	C₅×#67⋊C₄	semidirect
313. C₃	C₅×#893	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₅×C₃⋊D₃	2
C₂×C₅×#626	2
C₂×C₅×D₃×Dic₃	2
C₃×C₅×(C₂×D₃)⋊C₄	2
C₃×C₅×Dic₃⋊C₄	2
C₅×#3722	2
C₅×Dic₃²	2
C₅×#893	3
C₅×#67⋊C₄	3
#63881	5

Subgroups — C5×#63881 #13087165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#63881 #13087165